Compact sets in a subspace versus compact sets in the entire space

Question

Is it true that the compact sets in a subspace $\Omega$ of a topological space $X$ include all the subsets of $\Omega$ which are compact in the entire space $X$? If not, what simple property should $\Omega$ satisfy for this to hold?

Answer 1

Yes, it is true. If $Y\subset\Omega$ and $Y$ is compact in $X$, let $(O_\lambda)_{\lambda\in\Lambda}$ be an open cover of $Y$ in $\Omega$. Then each $O_\lambda$ can be written as $A_\lambda\cap\Omega$, for some open set $A_\lambda\subset X$. But then $(A_\lambda)_{\lambda\in\Lambda}$ is an open cover of $Y$ and therefore it has an open subcover $(A_{\lambda_i})_{i=1}^n$. Therefore, $(O_{\lambda_i})_{i=1}^n$ is a finite open subcover of the cover $(O_\lambda)_{\lambda\in\Lambda}$.